Tuesday, October 25, 2016

notes for a spirolateral bestiary

The simple rules for creating generalized spirolaterals produce a surprisingly diverse menagerie of paths which trace out some familiar and less-familiar shapes, including regular polygons, star polygons, tangles, wreaths and infinitely long springs.

some spirolaterals (graph here)

In the formula below, a represents the angle that you turn by at each step, and m represents the maximum length of the sides that you count up to before repeating. If m = 1, all sides are length 1, if m = 2, then the sides alternate between length 1 and length 2, for m = 3, the lengths of the sides form the repeating squence 1, 2, 3, 1, 2, 3... etc.
If the angle that you turn by is a rational multiple of 2pi, then the first spirolateral (m = 1) will trace out either a regular polygon or a regular star polygon - it is always adding a length of 1, and since the angle is a multiple of 2pi you will close the loop and end up back where you started.
Different story if the angle not a rational multiple of 2pi: you will never get back to the starting point, and the first spirolateral is going to look like an annulus after enough iterations (below is a = 2, zoomed in on the left, zoomed out on the right):

spirolateral that does not (soon) meet up - graph here

For higher values of m, although the shapes traced out vary widely they are always equiangular (we are always turning by the same angle, after all). So for m = 2 we end up with isogonal figures: every vertex is the same - it always has the same angle and always has sides of lengths 1 and 2 on either side of it. Here are some isogonal polygons - the first, the rectangle, is pretty familiar; the second, @solvemymaths tells us, may be a ditrigon.

And here are a few star-isogonals:

What about the beasties that spiral off to infinity? One set of these springs occur when a = pi/k for some positive integer k  and m = 2k.

And then there are the tangled stars... not sure where to begin with these.

Trying classify even a few of these strange creatures reminds me of Borges' Book of Imaginary Beings.